\[\theta_{2}\left(z\middle|\tau\right)=\theta_{1}\left(z+\tfrac{1}{2}\pi\middle|%
\tau\right)=M\theta_{3}\left(z+\tfrac{1}{2}\pi\tau\middle|\tau\right)=M\theta_%
{4}\left(z+\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\tau\middle|\tau\right),\]